Free energy landscapes in spherical spin glasses (Q6566367)

The author introduces and analyzes free energy landscapes defined by associating to any point inside a suitable sphere a free energy calculated on a thin spherical band around it, using many orthogonal replicas. Based on such a concept for general spherical models several results related to the Thouless-Anderson-Palmer (TAP) approach, originally introduced for the Sherrington-Kirkpatrick model, are proved. The author establishes a TAP representation for the free energy, valid for any overlap value which can be sampled as many times as one wishes (multisamplable overlaps) in an appropriate sense.\N\NThe correction to the Hamiltonian in the TAP representation arises as the free energy of a certain model on an overlap dependent band. For the largest multi-samplable overlap it coincides with the Onsager reaction term, for smaller multi-samplable overlaps the obtained formula is new. The corresponding TAP equations for critical points are also derived. The proof does not appeal to the celebrated Parisi formula or the ultrametricity property.\N\NIt is proved that any overlap value in the support of the Parisi measure is multisamplable. For generic models, it is further shown that the set of multi-samplable overlaps coincides with a certain set that arises in the characterization for the Parisi measure by Talagrand. The author shows that the ultrametric tree of pure states can be embedded in the interior of the sphere in a natural way and, for this embedding, proves that the points on the tree uniformly maximize the free energies. From this one concludes that the Hamiltonian at each point on the tree is approximately maximal over the sphere of same radius, and that points on the tree approximately solve the TAP equations for critical points.